How to color particles based on the texture of the object from which they are emitted, using the "grid" emission setting?
A bit 'complicated' solution, as it seems that the particles UV are relative to the mesh boundings when the 'grid' option is set, not to the parent UV map (so I don't think this is possible to tune the nodes to obtain the wanted result, if you want to use particles) :



To do it :


Tune your particle system to obtain the grid dimensions you need
Apply the particle system, by clicking convert here :





You will obtain a duplicate of the cubes (on the right here) :





This cubes are dupli from the same object, so make them single user, by selecting them all and use the "make single user/object &amp; data' menu, like below, and confirm 'selected objects'





Now join them all in order to have 1 mesh from them. As they are still all selected use Ctrl+J
Enter edit mode Tab
Them select all the vertices and unwrap U using the 'smart UV project' mode. It will give you some kind of UV grid like in the right part of the picture below :




Now we will make a bake from the original to the cube shape. This bake wont work directly because the proximity between the cubes and the original mesh is not good.
So, to overpass that :


Add a shrinkwrap modifier to the cubes and set your original mesh as target object :




Now we can bake :


1 : Add a material to your cube mesh
2 : Add a image texture with a newly created texture. Keep this node active.
3 : Select your original mesh RMB then the cube mesh with Shift+RMB 
4 : Go to the bake part of the render panel and check 'selected to active'
5 : click 'bake'





Disable or remove the shrinkwrap modifier we set above
Using the baked texture, you obtain this below, with several colors on each cube :




If you need one color per cube, (edit : one color per face !)


Go to the UV editor
Select all your UVs
Set pivot to individual origins
Then scale to 0 S0




Doing that each cube UV is reduced to one point, and we obtain this :



If needed you can separate the cube in edit mode, using P then 'by loose parts' (this may take some calculations...).